Variational derivation of governing differential equations for truncated version of Bresse-Timoshenko beams I. Elishakoff a,n , F. Hache a,b , N. Challamel b a Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33481-0991, USA b Université de Bretagne Sud UBS, University of South Brittany, FRE CNRS 3744 IRDL , Centre de Recherche, Rue de Saint Maudé, BP92116, 56321 Lorient Cedex, France article info Article history: Received 7 March 2017 Received in revised form 19 July 2017 Accepted 21 July 2017 Handling editor: W. Lacarbonara Keywords: Bresse-Timoshenko beams Natural frequencies Asymptotic derivation Variational derivation Truncated version Slope-inertia version abstract In this paper, we provide a variational derivation of modified Bresse-Timoshenko equa- tions. This process leads to the additional term to so called truncated set of Bresse-Ti- moshenko equations; this new term is associated with the modified slope inertia. The two sets of truncated versions of the Bresse-Timoshenko equations are contrasted with each other, as well as with the original Bresse-Timoshenko equations on the example of the (a) beam that is simply supported at its both ends, and (b) the cantilever beam with end mass. It is concluded that application of either truncated sets of equations is advantageous over the original Bresse-Timoshenko equations. As far as two truncated sets of equations are concerned, the variationally derived set appears to be preferable. & 2017 Published by Elsevier Ltd. 1. Introduction Timoshenko [1] introduced his governing differential equations that take into account shear deformation and rotary inertia. As far as the effect of rotary inertia is concerned Timoshenko [1] had two predecessors, namely Bresse [2] and Rayleigh [3]. Unfortunately, he did not make a reference to either of them. Neither Rayleigh [3] made a reference to the work of Bresse [2]. Probably these facts led Koiter [4] to remark: “What is generally known as Timoshenko beam theory is a good example of a basic principle in the history of science: a theory which bears someone’s name is most likely due to someone else”. In these circumstances, it appears justified to refer to the beam theory that takes into account the rotary inertia and shear deformation as Bresse-Timoshenko beam theory. Extensive review of Bresse-Timoshenko theory was provided re- cently by Elishakoff et al. [5]. Timoshenko [1] derived his governing equations both by equilibrium and variational method for small deflections and small angles. This model has been widely used in the literature and leads to an equation in displacement with a fourth order time derivative. In their work on this topic, Weaver et al. [6] evaluated the relative contribution of various terms in characteristic equation associated with simply supported beam, in his equations. They arrived at the following conclusion that this term was extremely small compare to the others. However, they fell short of suggesting to neglect the fourth order derivative Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration http://dx.doi.org/10.1016/j.jsv.2017.07.039 0022-460X/& 2017 Published by Elsevier Ltd. n Corresponding author. E-mail addresses: elishako@fau.edu (I. Elishakoff), fhache2014@fau.edu (F. Hache), noel.challamel@univ-ubs.fr (N. Challamel). Journal of Sound and Vibration 409 (2017) ∎∎∎–∎∎∎ Please cite this article as: I. Elishakoff, et al., Variational derivation of governing differential equations for truncated version of Bresse-Timoshenko beams, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.07.039i